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Created by Michael Jones

A shock (also called a shock wave) is a coalescence of pressure waves in the supersonic regime. A shock causes a sudden change to the properties of a flow in a very short distance.

## Types of ShocksEdit

### Normal ShockEdit

A normal shock occurs when a shock forms perpendicular to the flow velocity. it usually occurs in pipe flow and can be either a moving or stationary wave. In a moving wave, the shock is moving at supersonic speeds through a stationary medium, whereas a stationary wave is fixed while a supersonic fluid moves through the wave. A normal shock effectively reduces the flow of a fluid from supersonic to subsonic velocities. A normal shock can occur at the leading edge of a body when the object is blunt and when the velocity is supersonic, but close to Mach 1, but transitions to an oblique shock at higher Mach numbers. It can also be observed on the upper surface of an airfoil if flying at transonic speeds above the critical Mach number.[2]

### Oblique ShockEdit

An oblique shock is oriented at some angle to the direction of the flow. It is usually observed at the leading edge of a sharp body or in supersonic flow through an overexpanded nozzle. An oblique shock can have either a strong or a weak shock solution based on the angle of the shock from horizontal, $\theta$. This angle changes also based on the angle of inclination of the body, $\delta$. When $\delta$ approaches zero, the strong solution tends towards a normal shock, while the weak solution becomes a Mach Wave, which is a series of pressure waves that are not as concentrated as a shock. The Mach number after an oblique shock is less than the original, but is usually still greater than 1. This is due to the fact that only the component of velocity that is normal to the shock is affected whereas the velocity component tangent to the shock remains the same. When the components are added together after the shock, the resulting velocity is oriented along the slope of the body at angle $\delta$. The effect of an oblique shock is not only to slow the flow, but also to change its direction.[2]

To see how various inputs change an oblique shock and its resulting flow, an interactive tool can be found on NASA's website.

### Bow ShockEdit

A bow shock is similar to an oblique shock in that they both form in front of a body in supersonic flow, but a bow shock is detached and forms a curve. Gererally, a bow shock occurs before a blunt object like the one shown to the right. With the object moving from right to left (conversely assuming a moving fluid from left to right), the shock acts like a normal shock at the vertical centerpoint because the flow is normal to the shock. Moving away from the centerpoint, the shock becomes more oblique. A detached bow shock forms in front of a body when the flow has to turn at a sharper angle than can be achieved with an oblique shock.[2][4]

## Properties of a Shock WaveEdit

When a fluid travels faster than the speed of sound (ie. supersonic), the effect of a disturbance cannot propagate upstream, so a shock acts as a way to alter the characteristics of a fluid and change its flow direction to avoid the disturbance. This occurs within a distance on the order of 10-5 meters.[4] In the image to the right, region 1 (upstream of the shock) is supersonic indicated by a Mach Number greater than 1. Across the shock, into region 2, the pressure, density, temperature and entropy increase while the Mach Number, velocity and total pressure decrease. The process of crossing a shock is adiabatic as there is no heat gain or loss, and therefore the enthalpy remains constant across a shock. In the normal shock, the Mach number after the shock is subsonic. Due to high viscous effects and internal heat conduction effects in the shock region increase entropy rendering this process non-isentropic.[2]

### Normal Shock RelationsEdit

The ratio of properties across a normal shock are known as the Normal Shock Relations. They take into account the velocity relative to the speed of sound and the ratio of specific heats, and are used to calculate the properties of the flow across a normal shock.[4]

The first relation solves for the Mach number after the shock (M2).

$M^{2}_{2}=\frac{1+(\frac{\gamma-1}{2})M^{2}_{1}}{\gamma M^{2}_{1}-(\frac{\gamma-1}{2})}$

The second relation gives the ratio of static pressure before and after the shock (p2/p1).

$\frac{p_{2}}{p_{1}}=1+\frac{2\gamma}{\gamma+1}(M^{2}_{1}-1)$

Next, we solve for the temperature ratio (T2/T1).

$\frac{T_{2}}{T_{1}}=[1+\frac{2\gamma}{\gamma+1}(M^{2}_{1}-1)]\frac{2+(\gamma-1)M^{2}_{1}}{(\gamma+1)M^{2}_{1}}$

Finally, we solve for the density ratio ($\rho_{2}/\rho_{1}$).

$\frac{\rho_{2}}{\rho_{1}}=\frac{M^{2}_{1}}{1+\frac{\gamma-1}{\gamma+1}(M^{2}_{1}-1)}$

Using these relations, we can determine the change in entropy of the system (s2-s1).

$s_{2}-s_{1}=c_{p}ln(\frac{T_{2}}{T_{1}})-Rln(\frac{p_{2}}{p_{1}})$

## Design ConsiderationsEdit

When designing an aircraft the possible occurrence of shocks play a role in the preliminary design. When flying at supersonic speeds, there must be a consideration for the effect of wave drag, or the component of total drag that is due to the viscous effects of the air passing through a shock. Airfoils need a sharp leading edge and engine inlets and exhausts must be sized and shaped correctly to minimize the losses from shocks.[2]

### Critical Mach NumberEdit

Even when flying at transonic speeds, the air flowing over an airfoil can reach supersonic speeds causing a normal shock to form and causing flow separation above the airfoil. The speed at which this occurs is referred to as the Critical Mach Number, and the drastic increase in drag associated with this shock is known as Drag Divergence. To reduce this effect, a supercritical airfoil may be used that decreases the camber along the majority of the wing but has a scoop on the underside of the trailing edge that effectively creates a high camber. This reduces the speed over the top of the wing to subsonic levels, eliminating the shock.[6]

### NoiseEdit

When an aircraft is flying at supersonic speeds the series of shocks around the airframe produce powerful pressure waves of a characteristic frequency commonly referred to as a sonic boom. These waves propagate through the air and are audible from the ground below. This issue is a primary design concern that limits the ability to produce supersonic vehicles that fly above populated areas due to governmental restrictions on noise pollution. The wave produced is called an N-wave that is characterized by a sharp increase in pressure that linearly decreases below ambient pressure and then another sharp increase back to ambient. The first rise is due to the shock wave of the nose and the second is due to the shocks from the tail. Current Federal Aviation Regulations (FARs) prohibit the flight of commercial aircraft above Mach 1. The flight of the Concorde was allowed to fly into the US providing that they do not "cause a sonic boom to reach the surface of the United States" (14 CFR Part 91).[9]

Several companies are researching ways to suppress the severity of the sonic boom by reducing the peaks on the N-Wave diagram. Gulfstream Aerospace is hoping to lessen the effect of the initial peak by using a graduated "spike" attached to the nose to create a series of smaller shocks.[7] Northrop Grumman has been testing how the smooth graduation of airframe cross-sectional area in the nose section of the F-5 can reduce the initial peak.[8]

While not eliminating sonic booms altogether, using these and other methods to reduce the sonic boom signature could lead to changes in FAA regulations that could allow the use of commercial supersonic vehicles for cross-continental flights.[9]

## Shock DiamondsEdit

 SR-71B during takeoff showing exhaust shock diamonds. NASA photo. Shock and expansion wave interaction in an overexpanded nozzle.[10] Shock diamonds in an underexpanded nozzle.[10]

The effects of shocks can be seen clearly in what are known as shock diamonds. In a rocket or afterburning jet engine with a contracting/expanding nozzle, the visible plume will appear to have brighter "diamonds" at regular intervals as seen in the photo of the SR-71 to the right. This is caused by the interaction of oblique shocks and Prandtl-Meyer Expansion waves. The flow is sonic (or "choked") at the smallest point of the nozzle and the following expansion accelerates the flow. Given a certain back-pressure or atmospheric pressure, there is a particular exit area that produces optimum efficiency. If, however, the pressure changes for example due to a change in altitude, the nozzle is considered over- or underexpanded.[2]

If overexpanded (ie. the atmospheric pressure is greater than the exhaust pressure), an oblique shock forms at the lip of the nozzle, turning the flow inward and increasing the pressure. If this pressure is greater than PA, a series of expansion waves accelerate the flow outward. This process continues until the pressures are equal. The same process occurs for an underexpanded nozzle when PA < PE, but the expansion waves turn the flow outward first.[2]

These two conditions can be seen in the diagrams to the right.

## ReferencesEdit

1. http://www.ae.gatech.edu/people/jseitzma/classes/ae3450/
2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 John, James E.A., Gas Dynamics. 2nd ed. 1984, Upper Saddle River, N.J.: Pearson Prentice Hall.
3. http://www.hq.nasa.gov/office/pao/History/SP-440/ch6-2.htm
4. 4.0 4.1 4.2 Anderson, J.D., Fundamentals of Aerodynamics. 3rd ed. McGraw-Hill Series in Aeronautical and Aerospace Engineering. 2001, Boston: McGraw-Hill.
5. United States Flight Standards Service. Airman Testing Standards Branch., Airplane flying handbook. 2004, Washington, D.C.: U.S. Dept. of Transportation, Federal Aviation Administration.
6. Anderson, John D., Aircraft Performance and Design. 1999, Boston: WCB/McGraw-Hill.
7. 7.0 7.1 Preston Henne et al., Gulfstream Aerospace, Supersonic Aircraft with Spike for Controlling and Reducing Sonic Boom, US Patent Application No.11/307280 (pending, continuation-in-part of US patent No. 6698684), filed 30 June 2006, published 28 June 20.
8. 8.0 8.1 http://www.popsci.com/military-aviation-space/article/2004-07/whooshhh
9. 9.0 9.1 National Research Council (U.S.). Committee on Breakthrough Technology for Commercial Supersonic Aircraft., Commercial supersonic technology : the way ahead. Compass series. 2001, Washington, D.C.: National Academy Press.
10. 10.0 10.1 http://www.aerospaceweb.org/question/propulsion/q0224.shtml